> [!tldr] > The **discrete-time Fourier transform (DTFT)** transforms a discrete sequence $(x_{n})_{n}$ (e.g. the values $f(n)$ of the function $f$ at all integers $n$) to a function given by $X(\omega):= \sum_{n}x_{n}\exp(-i\omega n),$defined for $\omega \in (0, 2\pi)$. The discreteness refers to the discreteness of inputs. In particular, if a function $x(z)$ on $\mathbb{R}$ is sampled at fixed discrete intervals ($\Delta T$) and the frequency is standardized with $\omega:= f / 2\pi \Delta T$, then the DTFT is a Riemann sum approximating the [[Continuous Fourier Transform|continuous FT]] of frequency $f$: $\sum_{n}\underbrace{\Delta T \cdot x(n\Delta T)}_{=: x_{n}} \cdot \exp(-2\pi in \Delta T \cdot \omega) \approx \int x(z)\exp(-2\pi izf) ~ dz. $